Answer

$x=-\frac{4}{3}$ or $x=3$ or $x=-3$

Work Step by Step

We solve by factoring with grouping:
$3x^{3}+4x^{2}=27x+36$
$3x^{3}+4x^{2}-27x-36=0$
$x^{2}(3x+4)-9(3x+4)=0$
$(3x+4)(x^{2}-9)=0$
$(3x+4)(x-3)(x+3)=0$
$3x+4=0$ or $x-3=0$ or $x+3=0$
$3x=-4$ or $x=3$ or $x=-3$
$x=-\frac{4}{3}$ or $x=3$ or $x=-3$